Harmonic multivector fields and the Cauchy integral decomposition in Clifford analysis

Abstract

In this paper we study the problem of decomposing a Hölder continuous $k$-grade multivector field $F_{k}$ on the boundary $\Gamma$ of an open bounded subset $\Omega$ in Euclidean space $\R^{n}$ into a sum $F_{k}=F_{k}^{+}+F_{k}^{-}$ of harmonic $k$-grade multivector fields $F_{k}^{\pm}$ in $\Omega_{+}=\Omega$ and $\Omega_{-}=\R^{n}\setminus (\Omega\cup\Gamma)$ respectively. The necessary and sufficient conditions upon $F_{k}$ we thus obtain complement those proved by Dyn'kin in [20,21] in the case where $F_{k}$ is a continuous $k$-form on $\Gamma$. Being obtained within the framework of Clifford analysis and hence being of a pure function theoretic nature, they once more illustrate the importance of the interplay between Clifford analysis and classical real harmonic analysis.